Currents and Coil Forces as Contour Integrals in Two-Dimensional Magnetic Fields

A general method is given for evaluating currents and coil forces in two-dimensional magnetic fields. When the field components HX and HY are known along a closed contour C in the field plane Z=X+iY and we define H≡HY+iHX, the total current within C is ∮ C HdZ/4πiand the total field force per unit length, F≡FY+iFX, acting on all the currents within C is − ∮ C H2dZ/8π. As an example, the current and coil force are worked out for the uniform field magnet formed by intersecting parallel cylindrical conductors and compared with prior results.